Internal Error Propagation in Explicit Runge--Kutta Methods

Internal Error Propagation in Explicit Runge-Kutta Methods

In practical computation with Runge–Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depen...

Continuous Explicit Runge-Kutta methods

Explicit Stabilized Runge-Kutta Methods

2-stage explicit total variation diminishing preserving Runge-Kutta methods

In this paper, we investigate the total variation diminishing property for a class of 2-stage explicit Rung-Kutta methods of order two (RK2) when applied to the numerical solution of special nonlinear initial value problems (IVPs) for (ODEs). Schemes preserving the essential physical property of diminishing total variation are of great importance in practice. Such schemes are free of spurious o...

Global Error versus Tolerance for Explicit Runge-Kutta Methods

Initial value solvers typically input a problem specification and an error tolerance, and output an approximate solution. Faced with this situation many users assume, or hope for, a linear relationship between the global error and the tolerance. In this paper we examine the potential for such 'tolerance proportionality' in existing explicit Runge-Kutta algorithms. We take account of recent deve...